Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
394 conjugacy classes of subgroups; only the 172 normal subgroups are shown.
This group has 394 conjugacy classes of subgroups — too many to draw. Only its 172 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22⋊C4 |
| commutator | C22 | C23×C4 |
| Frattini, cosocle | C23 | C24 |
| socle | C22 | C2×C22⋊C4 |
| prime radical | C22×#41 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#117 | C2 | semidirect |
| 3. #410 | C2 | direct (N × H) |
| 4. C2×#99 | C2 | semidirect |
| 8. C2×#41 | C4 | semidirect |
| 10. C22×Q8 | C4 | semidirect |
| 11. C22×D4 | C4 | semidirect |
| 13. C23×C4 | C4 | semidirect |
| 16. #117 | C22 | semidirect |
| 20. #99 | C22 | semidirect |
| 29. C2×Q8 | C2×C4 | semidirect |
| 35. C2×D4 | C2×C4 | semidirect |
| 45. C22×C4 | C2×C4 | semidirect |
| 110. C2 | #410 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#41 | 2 |
| C2×#117 | 2 |
| #410 | 2 |
| C2×#99 | 2 |